Prove2Me

`BookProof.QuantumGravityBrstCharge.qgBRST_nilpotent` (M : Fin 19 → Matrix (Fin 84) (Fin 84) ℝ) (f : Fin 19 → Fin 19 → Fin 19 → ℝ) (hclose : ∀ a b, M a * M b - M b * M a = ∑ e, f a

Proved
BookProof.QuantumGravityBrstCharge.qgBRST_nilpotent

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterQuantumGravityBrstCharge.

BookProof.QuantumGravityBrstCharge.qgBRST_nilpotent (M : Fin 19 → Matrix (Fin 84) (Fin 84) ℝ) (f : Fin 19 → Fin 19 → Fin 19 → ℝ) (hclose : ∀ a b, M a * M b - M b * M a = ∑ e, f a b e • M e) (hf12 : ∀ a b c, f a b c = -f b a c) (hjac : ∀ a b c h : Fin 19, ∑ e, (f a b e * f e c h + f b c e * f e a h + f c a e * f e b h) = 0) : qgBRST M f * qgBRST M f = 0

Formalization note: Lean 4 identifier BookProof.QuantumGravityBrstCharge.qgBRST_nilpotent.

Preamble
-- Generated from ChapterQuantumGravityBrstCharge.lean — theorem BookProof.QuantumGravityBrstCharge.qgBRST_nilpotent
import Definitions.Def_ChapterBRSTNilpotent
import Definitions.Def_ChapterYangMillsHermite
import Definitions.Def_ChapterQuantumGravity3DGauge
import Mathlib
import Definitions.Def_ChapterQuantumGravityBrstCharge
open BookProof.QuantumGravityBrstCharge



open MvPolynomial BookProof.BRSTNilpotent BookProof.YangMillsHermite
open BookProof.QuantumGravity3DGauge

noncomputable section

variable {R : Type*} [Ring R] [Algebra ℝ R] {n : ℕ}
variable {f : Fin n → Fin n → Fin n → ℝ} {G χ β : Fin n → R}
variable {d : ℕ}
variable {α : Type*}
Formal statement
theorem BookProof.QuantumGravityBrstCharge.qgBRST_nilpotent (M : Fin 19 → Matrix (Fin 84) (Fin 84) ℝ)
    (f : Fin 19 → Fin 19 → Fin 19 → ℝ)
    (hclose : ∀ a b, M a * M b - M b * M a = ∑ e, f a b e • M e)
    (hf12 : ∀ a b c, f a b c = -f b a c)
    (hjac : ∀ a b c h : Fin 19,
      ∑ e, (f a b e * f e c h + f b c e * f e a h + f c a e * f e b h) = 0) :
    qgBRST M f * qgBRST M f = 0 := by sorry

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